An Inexact Cayley Transform Method For Inverse Eigenvalue Problems Zheng-Jian Bai¤ Raymond H. Chan¤ Benedetta Moriniy Abstract The Cayley transform method is a Newton-like method for solving inverse eigenvalue problems. If the problem is large, one can solve the Jacobian equation by iterative methods. However, iterative methods usually oversolve the problem in the sense that they require far more (inner) iterations than is required for the convergence of the Newton (outer) iterations. In this paper, we develop an inexact version of the Cayley transform method. Our method can reduce the oversolving problem and improves the e±ciency with respect to the exact version. We show that the convergence rate of our method is superlinear and that a good tradeo® between the required inner and outer iterations can be obtained. Keywords. Nonlinear equation, inverse eigenvalue problem, Cayley transform AMS subject classi¯cations. 65F18, 65F10, 65F15. 1 Introduction Inverse eigenvalue problems arise in a variety of applications, see for instances the pole assignment problem [5, 32], the inverse Toeplitz eigenvalue problem [8, 31, 35], the inverse Sturm-Liouville problem [1, 21], and also problems in applied mechanics and structure de- sign [18, 19, 22], applied geophysics [30], applied physics [23], numerical analysis [27], and dynamics systems [14]. A good reference for these applications is the recent survey paper on structured inverse eigenvalue problems by Chu and Golub [10]. In many of these appli- cations, the problem size n can be large. For example, large Toeplitz eigenvalue problems have been considered in [31]. Moreover, in the discrete inverse Sturm-Liouville problem, n is the number of grid-points, see Chu and Golub [10, p. 10]. Our goal in this paper is to derive an e±cient algorithm for solving inverse eigenvalue problems when n is large. n Let us ¯rst de¯ne the notations. Let fAkgk=0 be n + 1 real symmetric n-by-n matrices. T n For any c = (c1; : : : ; cn) 2 R , let Xn A(c) ´ A0 + ciAi; (1) i=1 ¤(zjbai, [email protected]) Department of Mathematics, Chinese University of Hong Kong, Shatin, NT, Hong Kong. The research of the second author was partially supported by the Hong Kong Research Grant Council Grant CUHK4243/01P and CUHK DAG 2060257. y(benedetta.morini@uni¯.it) Dipartimento di Energetica `S. Stecco' Universit`adi Firenze, Via C. Lom- broso 6/17, 50134 Firenze. Research was partially supported by MIUR, Rome, Italy, through \Co¯nanzia- menti Programmi di Ricerca Scienti¯ca di Interesse Nazionale". 1 n and denote the eigenvalues of A(c) by f¸i(c)gi=1, where ¸1(c) · ¸2(c) · ¢ ¢ ¢ · ¸n(c). An ¤ ¤ inverse eigenvalue problem (IEP) is de¯ned as follows: Given n real numbers ¸1 · ¢ ¢ ¢ · ¸n, n ¤ ¯nd c 2 R such that ¸i(c) = ¸i for i = 1; : : : ; n. We note that the IEP can be formulated as a system of nonlinear equations ¤ ¤ T f(c) ´ (¸1(c) ¡ ¸1; : : : ; ¸n(c) ¡ ¸n) = 0: (2) It is easy to see that a direct application of Newton method to (2) requires the computation of ¸i(c) at each iteration. To overcome the drawback, di®erent Newton-like methods for solving (2) are given in [17]. One of these methods, Method III, forms an approximate Jacobian equation by applying matrix exponentials and Cayley transforms. As noted in [7], the method is particularly interesting and it has been used or cited in [8, 9, 25, 33] for instances. If (2) is solved by Newton-like methods, then in each Newton iteration (the outer itera- tion), we need to solve the approximate Jacobian equation. When n is large, solving such a linear system will be costly. The cost can be reduced by using iterative methods (the inner iterations). Although iterative methods can reduce the complexity, they may oversolve the approximate Jacobian equation in the sense that the last tens or hundreds inner iterations before convergence may not improve the convergence of the outer Newton iterations [13]. In order to alleviate the oversolving problem, we propose in this paper an inexact Newton-like method for solving the nonlinear system (2). The inexact Newton-like method is a method that stops the inner iterations before convergence. By choosing suitable stopping criteria, we can minimize the oversolving problem and therefore reduce the total cost of the whole inner-outer iterations. In essence, one does not need to solve the approximate Jacobian equation exactly in order that the Newton method converges fast. In this paper, we give an inexact version of Method III where the approximate Jacobian equation is solved inexactly by stopping the inner iterations before convergence. We propose a new criterion to stop the inner iterations at each Newton step and provide theoretical and experimental results for the procedure. First, we will show that the convergence rate of our method is superlinear. Then, we illustrate by numerical examples that it can avoid the oversolving problem and thereby reduce the total cost of the inner-outer iterations. We remark that our proposed method is locally convergent. Thus, how to select the initial guess becomes a crucial problem. However, global continuous methods such as the homotopy method can be used in conjunction with our procedure. In these continuous methods, our inexact method can be used as the corrector step where a valid starting point is provided by the globalization strategy, see for examples [3] and [37, pp. 256{262]. This paper is organized as follows. In x2, we recall Method III for solving the IEP. In x3, we introduce our inexact method. In x4, we give the convergence analysis of our method. In x5, we present numerical tests to illustrate our results. 2 The Cayley Transform Method Method III in [17] is based on Cayley transforms. In this section, we briefly recall this ¤ method. Let c be a solution to the IEP. Then there exists an orthogonal matrix Q¤ satisfying T ¤ ¤ ¤ Q¤ A(c )Q¤ = ¤¤; ¤¤ = diag(¸1; : : : ; ¸n): (3) k ¤ Suppose that c and Qk are the current approximations of c and Q¤ in (3) respectively Zk T and that Qk is an orthogonal matrix. De¯ne e ´ Qk Q¤. Then Zk is a skew-symmetric 2 matrix and (3) can be written as 1 1 QT A(c¤)Q = eZk ¤ e¡Zk = (I + Z + (Z )2 + ¢ ¢ ¢ )¤ (I ¡ Z + (Z )2 + ¢ ¢ ¢ ): k k ¤ k 2 k ¤ k 2 k T ¤ 2 Thus Qk A(c )Qk = ¤¤ + Zk¤¤ ¡ ¤¤Zk + O(kZkk ), where k ¢ k denotes the 2-norm. k In Method III, c is updated by neglecting the second order terms in Zk, i.e. T k+1 Qk A(c )Qk = ¤¤ + Zk¤¤ ¡ ¤¤Zk: (4) We ¯nd ck+1 by equating the diagonal elements in (4), i.e. ck+1 is given by k T k+1 k ¤ (qi ) A(c )qi = ¸i ; i = 1; : : : ; n; (5) k n where fqi gi=1 are the column vectors of Qk. By (1), (5) can be rewritten as a linear system J (k)ck+1 = ¸¤ ¡ b(k); (6) ¤ ¤ ¤ T where ¸ ´ (¸1; : : : ; ¸n) , and h i (k) k T k J = (qi ) Ajqi ; i; j = 1; : : : ; n; (7) ij (k) k T k [b ]i = (qi ) A0qi ; i = 1; : : : ; n: (8) k+1 Once we get c from (6), we obtain Zk by equating the o®-diagonal elements in (4), i.e. k T k+1 k (qi ) A(c )qj [Zk]ij = ¤ ¤ ; 1 · i 6= j · n: (9) ¸j ¡ ¸i Finally we update Qk by setting Qk+1 = QkUk, where Uk is an orthogonal matrix con- structed by the Cayley transform for eZk , i.e. 1 1 U = (I + Z )(I ¡ Z )¡1: k 2 k 2 k We summarize the algorithm here. Algorithm I: Cayley Transform Method 0 0 n 0 1. Given c , compute the orthonormal eigenvectors fqi(c )gi=1 of A(c ). Let Q0 = 0 0 0 0 [q1;:::; qn] = [q1(c );:::; qn(c )]. 2. For k = 0; 1; 2;:::, until convergence, do: (a) Form the approximate Jacobian matrix J (k) by (7) and b(k) by (8). (b) Solve ck+1 from the approximate Jacobian equation (6). (c) Form the skew-symmetric matrix Zk by (9). k+1 k+1 k+1 k+1 T (d) Compute Qk+1 = [q1 ;:::; qn ] = [w1 ;:::; wn ] by solving 1 (I + Z )wk+1 = gk; j = 1; ¢ ¢ ¢ ; n; (10) 2 k j j k 1 T where gj is the jth column of Gk = (I ¡ 2 Zk)Qk . 3 This method was proved to converge quadratically in [17]. Note that in each outer iteration (i.e. Step 2), we have to solve the linear systems (6) and (10). When the systems are large, we may reduce the computational cost by solving both systems iteratively. One could expect that it requires only a few iterations to solve (10) iteratively. This is due to k ¤ the fact that, as fc g converges to c , kZkk converges to zero, see [17, Equation (3.64)]. Consequently, the coe±cient matrix on the left hand side of (10) approaches the identity matrix in the limit, and therefore (10) can be solved e±ciently by iterative methods. On the other hand, iterative methods may oversolve the approximate Jacobian equation (6), in the sense that for each outer Newton iteration, the last few inner iterations may not contribute much to the convergence of the outer iterations. How to stop the inner iterations e±ciently is the focus of our next section. 3 The Inexact Cayley Transform Method The main aim of this paper is to propose an e±cient version of Algorithm I for large problems. To reduce the computational cost, we solve both (6) and (10) iteratively with (6) being solved inexactly.